It is well know that seismic reflection amplitudes vary with the incidence angle or the acquisition offset. This variation in the seismic amplitude is governed by subsurface rock properties, such as P-wave velocity or impedance, S-wave velocity or impedance, and density. It is a fairly common practice in the geophysics community to predict or invert for these subsurface properties by exploiting this property of the seismic reflection amplitudes. Such inversion methodologies are conventionally referred to as Amplitude-vs-Angle (AVA) or Amplitude-vs-Offset (AVO) inversion. The most common types of reflection modes used in the geophysics community are PP and PS modes. A reflected PP mode is excited when both the incident and the reflected waves are P-wave, whereas a PS mode is defined by an incident P-wave but a reflected S-wave.
Reflectivity at the interface between two elastic media can be calculated by Zoeppritz's equations. However, Zoeppritz's equations are complex and, in their original form, do not provide any insight to the physics of the wave propagation. Due to their nonlinear form, they are difficult to use in linear inverse problems and may cause nonuniqueness in the inversion solutions. Hence, a number of its linearized forms have been proposed in the last thirty years for both isotropic and anisotropic media. Aki and Richard [1] derived linearized forms of reflectivity between two isotropic media for all possible modes of wave propagation. Their equations have been further simplified by several authors and have served as a basis for most of the AVA-based inversion.
As known by those of ordinary skill in the art, if the medium is isotropic or has polar anisotropy, the seismic reflection amplitudes vary only with the incidence angle. However, if the medium is azimuthally anisotropic, the seismic amplitudes vary both with the incidence angle and the azimuth of the incidence plane. It is well known that at seismic wavelengths, fractured reservoirs exhibit azimuthal anisotropy. More specifically, one set of parallel vertical fractures in isotropic rocks causes HTI anisotropy. Ruger and Tsvankin [2] proposed a seismic inversion method for HTI media. Their method, however, is limited to PP data only. They derived new linearized equations for the reflection coefficients for HTI media and they showed that the reflectivity between two HTI media is governed by the additional anisotropic or Thomsen's parameter [3] along with P-wave velocity and density.
S-waves while travelling through fractured rocks split into the fast (S1) and slow (S2) S-waves. This phenomenon, along with the anisotropic nature of reflection coefficients, compounds the problem of seismic inversion of PS modes. Due to azimuthally varying elastic properties in HTI media, seismic reflectivity also varies with azimuth of wave propagation. The extra anisotropic constants in HTI media and complex wave propagation render the equations for reflection coefficients very complex and cause the AVA-based inverse problem to be intractable. Due to these problems, inversion for PS data in azimuthally anisotropic media is not very popular.
Jilek [4] proposed a method to invert for isotropic and anisotropic subsurface parameters in azimuthally anisotropic media. His method simultaneously inverts for both isotropic and anisotropic parameters, which render his approach unstable and unsuitable for application on field data which usually have low signal-to-noise ratio.
Thus, there is a need for improvement in this field.